Range

Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain

Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain

Does anyone know a good way of finding the range of a function...I usually either man it out and look for the wholes and check its horizontal endpoints...or if its a simple function I find its inverse and check that functions domain

Here is another way. The range of a function are all so that the equation has a solution in the domain.

Here is another way. The range of a function are all so that the equation has a solution in the domain.

Yeah that is how I check the possibility for a function to not have an element of the range...but it dose not help me for finding the range...for example I could use that method to check whether or not 1 is in the range by saying and seeing if there is an x value that satisfies that equation...but it doesnt help me to go o well since then the range is this

Yeah that is how I check the possibility for a function to not have an element of the range...but it dose not help me for finding the range...for example I could use that method to check whether or not 1 is in the range by saying and seeing if there is an x value that satisfies that equation...but it doesnt help me to go o well since then the range is this

Here is an example. Let , the domain is of course all numbers. The range are all so that we can solve the equation for in the domain. We begin by writing, but that can only have a solution if (because the exponentinal is positive). Which means if then . Now to solve this equation we require . For those values we can solve the equation.